Hyers–Ulam stability of functional inequalities: a fixed point approach

Hyers’ theorem was generalized by Aoki [1] for additive mappings and by Rassias [28] for linear mappings by considering an unbounded Cauchy difference. A generalization of Rassias’ theorem was given by Gavruta [13] by replacing the unbounded Cauchy difference with a general control function. In 1982, Rassias [24] after the innovative approach of the Rassias’ theorem [28] replaced ‖x‖p + ‖y‖p by ‖x‖p · ‖y‖q for p, q ∈ R with p + q = 1. A generalization of Hyers–Ulam stability problem for the quadratic functional equation

was given by Skof [29] for mappings f : X → Y , where X is a normed space and Y is a Banach space. Cholewa [10] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an abelian group. Czerwik [11] proved the Hyers-Ulam stability of the quadratic functional equation. The stability problem of functional equations has been discussed by many mathematicians using different spaces and mappings. Park and Najati [22] proved the Hyers-Ulam stability of functional equations in real Banach spaces. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [2,3,5,16,21,24]).
In [18], Jun and Kim considered the following cubic functional equation: It is easy to show that the function f (x) = x 3 satisfies the functional equation (1.1) and every solution of the cubic functional equation is said to be a cubic mapping. Rassias [25] first introduced and investigated the quartic functional equation.
We recall a fundamental result in fixed point theory. For some recent papers on fixed point theory, see [4,6,14,19]. 7,12]) Let (U, d) be a complete generalized metric space and J : U → U be a strictly contractive mapping with Lipschitz constant L < 1. Then, for each given element x ∈ U, either d J n x, J n+1 x = ∞ for all nonnegative integers n or there is a positive integer n 0 such that (1) d(J n x, J n+1 x) < ∞ for all n ≥ n 0 ; (2) the sequence {J n x} converges to a fixed point y * of J; (3) y * is the unique fixed point of J in the set Y = {y ∈ U|d(J n 0 x, y) < ∞}; (4) d(y, y * ) ≤ 1 1-L d(y, Jy) for all y ∈ Y .
We will use the following notations: • M n (U) is the set of all n × n-matrices in U; • e j M 1,n (C) means that jth component is 1 and the other components are zero; • E ij M n (C) is that the (i, j)-component is 1 and the other components are zero; • E ij ⊗ x M n (C) means that the (i, j)-component is x and the other components are zero; • for x M n (U), y M n (U), Note that (U, · n ) is a matrix normed space if and only if (M n (U), · n ) is a normed space for each positive integer n and AxB k ≤ A B x n holds for A ∈ M n (C), x = [x ij ] ∈ M n (C) and B ∈ M n,k (C)) and that (U, · n ) is a matrix Banach space if and only if U is a Banach space and (U, · n ) is a matrix normed space. A matrix Banach space (U, · n ) is called a matrix Banach algebra if U is an algebra.
Let E, F be vector spaces. For a given mapping h : E → F and a given positive integer n, This paper is organized as follows: In Sects. 2 and 3, using the fixed point method, we prove the Hyers-Ulam stability of the cubic and quartic functional equation in matrix Banach algebras. In Sects. 4 and 5, using the fixed point method, we prove the Hyers-Ulam stability of the additive and quartic functional equation in matrix Banach algebras. In 1996, Rassias and Isac [17] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using the fixed point method, the stability problems of several functional equations have been extensively investigated by a number of authors (see [9,23]).
Throughout this paper, we assume that X is a matrix normed space and that Y is a matrix Banach algebra.

Fixed points and Hyers-Ulam stability of a cubic and quartic functional equation: an even case
One can easily show that an even mapping f : X → Y satisfies (1.3) if and only if the even mapping f : X → Y is a quartic mapping, i.e., and that an odd mapping f : It is easy to show that the function f (x) = ax 3 + bx 4 satisfies the functional equation (1.3).
For a given mapping f : for all x, y ∈ X. Using the fixed point method, we prove the Hyers-Ulam stability of the functional equation Df (x, y) = 0 in matrix Banach algebras: an even case.
Proof Setting n = 1 in (2.2), we get for all x ∈ X. Consider the set and introduce the generalized metric on S: It is easy to show that (S, d) is complete (see [8,Theorem 2.5]). Now we consider the linear mapping J : S → S such that for all x ∈ X. Then g : X → Y is an even mapping. It follows from (2.5) and (2.6) that for all x ∈ X. Hence d(g, Jg) ≤ L 32 . By Theorem 1.2, there exists a mapping Q : X → Y satisfying the following: for all x ∈ X. Then Q : X → Y is an even mapping. The mapping Q is a unique fixed point of J in the set This implies that Q is a unique mapping satisfying (2.7) such that there exists a K ∈ (0, ∞) satisfying for all x ∈ X.
(2) d(J n g, Q) → 0 as m → ∞. This implies the equality for all x ∈ X. Jg), which implies the inequality for all x, y ∈ X. So DQ(x, y) = 0 for all x, y ∈ X. Since Q : X → Y is even, the mapping Q : X → Y is a quartic mapping. By Lemma 1.3, there exists a unique quartic mapping Q : X → Y satisfying (2.3), as desired.
where ζ > 0 is a constant. Define a function f q : R → R by Then f q satisfies the functional inequality for all x ∈ R. Let for all x ∈ R. We define the set S = {g q : R → R, g q (0) = 0} and consider the generalized metric on S as described in the proof of the above theorem. Also consider the mapping J : S → S such that . It is clear that Moreover, we have Also we can show that Jg).
The above result implies the following: Similarly, one can obtain a similar result to Corollary 2.3: Let 0 < p < 4 and φ ≥ 0 be real numbers and f : X → Y be a mapping satisfying (2.10). Then there exists a unique quartic

Fixed points and Hyers-Ulam stability of a cubic and quartic functional equation: an odd case
Using the fixed point method, we prove the Hyers-Ulam stability of the functional equation Df (x, y) = 0 in matrix Banach algebras: an odd case.
Proof Setting n = 1 in (2.2), we get for all x ∈ X. Replacing x by -x in (3.4), we get for all x ∈ X. Consider the set S := g : X → Y , g(0) = 0 and introduce the generalized metric on S: It is easy to show that (S, d) is complete (see [8,Theorem 2.5] for all g, h ∈ S. for all x ∈ X. Then g : X → Y is an odd mapping. It follows from (3.3) and (3.4) that for all x ∈ X. Hence d(g, Jg) ≤ L 16 . By Theorem 1.2, there exists a mapping C : X → Y satisfying the following: (1) C is a fixed point of J, i.e., for all x ∈ X. Then C : X → Y is an odd mapping. The mapping C is a unique fixed point of J in the set This implies that C is a unique mapping satisfying (3.6) such that there exists a K ∈ (0, ∞) satisfying for all x ∈ X.
(2) d(J n g, C) → 0 as m → ∞. This implies the equality for all x ∈ X.
for all x, y ∈ X. So DC(x, y) = 0 for all x, y ∈ X. Since C : X → Y is odd, the mapping C : X → Y is a cubic mapping. By Lemma 1.3, there exists a unique cubic mapping C : X → Y satisfying (3.2), as desired.
Corollary 3.2 Let p > 3 and φ ≥ 0 be real numbers and f : X → Y be a mapping satisfying (2.10). Then there exists a unique cubic mapping C : X → Y satisfying Proof The proof follows from Theorem 3.1 by taking L = 2 3-p and for all x, y ∈ X.
Combining Corollaries 2.3 and 3.2, we get the following.
Similarly, one can obtain a similar result to Corollary 3.2: Let 0 < p < 3 and φ ≥ 0 be real numbers and f : X → Y be a mapping satisfying (2.10). Then there exists a unique cubic mapping C : X → Y satisfying Combining Remarks 2.4 and 3.4, we get the following. Theorem 3.5 Let 0 < p < 3 and φ ≥ 0 be real numbers and f : X → Y be a mapping satisfying (2.10). Then there exist a unique quartic mapping Q : X → Y and a unique cubic mapping C : X → Y satisfying

Fixed points and Hyers-Ulam stability of an additive and quartic functional equation: an even case
One can easily show that an even mapping f : X → Y satisfies (1.4) if and only if the even mapping f : X → Y is a quartic mapping, i.e., and that an odd mapping f : X → Y satisfies (1.4) if and only if the odd mapping f : X → Y is an additive mapping, i.e., It is easy to show that the function f (x) = ax + bx 4 satisfies the functional equation (1.4).
For a given mapping f : X → Y , we define (-y) for all x, y ∈ X. Using the fixed point method, we prove the Hyers-Ulam stability of the functional equation Cf (x, y) = 0 in matrix Banach algebras: an even case.  ϕ(x ij , 0) + ϕ(-x ij , 0) (5.1) [x ij ] ∈ M n (X).